What is the general solution for 2sin(2x)+6sin(x)-2cos(x)=3 ?

The general solution for 2sin(2x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin

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2sin(2x)+6sin(x)-2cos(x)=3

2 sin (2 x) + 6 sin (x) - 2 cos (x) = 3

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

2 sin (2 x) + 6 sin (x) - 2 cos (x) = 3

Subtract

3

from both sides

2 sin (2 x) + 6 sin (x) - 2 cos (x) - 3 = 0

Rewrite using trig identities

- 3 - 2 cos (x) + 6 sin (x) + 4 cos (x) sin (x) = 0

Factor

- 3 - 2 cos (x) + 6 sin (x) + 4 cos (x) sin (x) : (2 cos (x) + 3) (2 sin (x) - 1)

(2 cos (x) + 3) (2 sin (x) - 1) = 0

Solving each part separately

2 cos (x) + 3 = 0 or 2 sin (x) - 1 = 0

2 cos (x) + 3 = 0 :

No Solution

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (2 x) = \frac{4}{5}

2 csc (x) + 17 = 15 + csc (x)

cos (x) - sin (x) = 2

cot (θ) = (5.8473)

6 = 2 + 5 arctan (\frac{x}{2})

What is the general solution for 2sin(2x)+6sin(x)-2cos(x)=3 ?
The general solution for 2sin(2x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin